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Numerical study of a family of dissipative KdV equations
1. | LAMFA, UMR 6140, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens, France, France |
References:
[1] |
M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems,, Communications on Pure and Applied Analysis, 7 (2008), 211.
|
[2] |
M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., (). Google Scholar |
[3] |
M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265.
doi: 10.1016/j.physd.2004.01.023. |
[4] |
C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes,, (French) [Multilevel schemes for waves equations], 27 (2009), 180.
doi: 10.1051/proc/2009027. |
[5] |
J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings,, Journal of Scientific Computing, 20 (2004), 159.
|
[6] |
J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, preprint, (2011). Google Scholar |
[7] |
D. Dutykh, Modélisation mathématique des Tsunamis,, (French) [Mathematical modeling of Tsunamis], (2007). Google Scholar |
[8] |
J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eq., 74 (1988), 369.
doi: 10.1016/0022-0396(88)90010-1. |
[9] |
J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356.
doi: 10.1006/jdeq.1994.1071. |
[10] |
O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.
|
[11] |
O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25.
doi: 10.1006/jdeq.2001.4163. |
[12] |
D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications,", SIAM Philadelphia, (1977).
doi: 10.1115/1.3424477. |
[13] |
S. Mallat, "A Wavelet Tour of Signal Processing,", Academic press, (1998).
|
[14] |
A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Cambridge University Press, (2005).
|
[15] |
L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, (). Google Scholar |
[16] |
E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of fluids, 12 (1969), 2388.
doi: 10.1063/1.1692358. |
[17] |
E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of fluids, 13 (1970), 1432.
doi: 10.1063/1.1693097. |
[18] |
A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation,, IMA J. Num. Anal., 20 (2000), 235.
doi: 10.1093/imanum/20.2.235. |
[19] |
R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, (1997).
doi: 10.1007/978-1-4684-0313-8. |
[20] |
S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcialaj Ekvacioj, 54 (2011), 119.
doi: 10.1619/fesi.54.119. |
[21] |
S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations,, Asymptot. Anal., 68 (2010), 155.
|
show all references
References:
[1] |
M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems,, Communications on Pure and Applied Analysis, 7 (2008), 211.
|
[2] |
M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., (). Google Scholar |
[3] |
M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265.
doi: 10.1016/j.physd.2004.01.023. |
[4] |
C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes,, (French) [Multilevel schemes for waves equations], 27 (2009), 180.
doi: 10.1051/proc/2009027. |
[5] |
J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings,, Journal of Scientific Computing, 20 (2004), 159.
|
[6] |
J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, preprint, (2011). Google Scholar |
[7] |
D. Dutykh, Modélisation mathématique des Tsunamis,, (French) [Mathematical modeling of Tsunamis], (2007). Google Scholar |
[8] |
J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eq., 74 (1988), 369.
doi: 10.1016/0022-0396(88)90010-1. |
[9] |
J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356.
doi: 10.1006/jdeq.1994.1071. |
[10] |
O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.
|
[11] |
O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25.
doi: 10.1006/jdeq.2001.4163. |
[12] |
D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications,", SIAM Philadelphia, (1977).
doi: 10.1115/1.3424477. |
[13] |
S. Mallat, "A Wavelet Tour of Signal Processing,", Academic press, (1998).
|
[14] |
A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Cambridge University Press, (2005).
|
[15] |
L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, (). Google Scholar |
[16] |
E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of fluids, 12 (1969), 2388.
doi: 10.1063/1.1692358. |
[17] |
E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of fluids, 13 (1970), 1432.
doi: 10.1063/1.1693097. |
[18] |
A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation,, IMA J. Num. Anal., 20 (2000), 235.
doi: 10.1093/imanum/20.2.235. |
[19] |
R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, (1997).
doi: 10.1007/978-1-4684-0313-8. |
[20] |
S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcialaj Ekvacioj, 54 (2011), 119.
doi: 10.1619/fesi.54.119. |
[21] |
S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations,, Asymptot. Anal., 68 (2010), 155.
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